Experiments in Fluids 30 (2001) 672±682 Ó Springer-Verlag 2001
Passive acoustic bubble sizing in sparged systems R. Manasseh, R. F. LaFontaine, J. Davy, I. Shepherd, Y.-G. Zhu
672 Abstract Passive acoustic bubble sizing was investigated in both controlled tests and in a stirred, sparged tank typical of the biotechnology or minerals processing industries. Acoustic techniques have promise for industrial systems where other bubble analysis methods are impractical. Acoustic signals were studied for bubbles precisely formed at higher air¯ow rates. Acoustic pulses varied with bubble production rate as well as with bubble size. A technique of windowing pulses is proposed. Two alternative versions of this windowing technique were applied to a stirred, sparged tank, giving good agreement. It was shown that, in some cases, it may also be possible to acoustically estimate the spatial distribution of void fraction.
1 Introduction Measurement of bubble-size distributions continues to be a problem for many systems. Bubbles of gas (often air) are introduced into many industrial and environmental ¯ows to promote a chemical or biological reaction. Aeration (or `sparging') systems are found in the chemical, minerals processing and pyrometallurgy industries, and in food, beverage and wastewater processing. In many cases, the bubble-size distribution is a vital control on the rate of gas-liquid mass transfer. In other cases, bubbles of a certain size are nuisances that must be removed. In the environment, entrapment of bubbles of varying sizes by breaking waves is a mechanism by which oceans absorb `greenhouse' gases. For testing industrial aeration equipment, standard void-fraction instruments are available. A sharp-tipped probe registers a binary change in conductivity or refractive index as it pierces a bubble. Clearly, the timeaveraged signal from such instruments will represent the
Received: 13 June 2000/Accepted: 19 October 2000
local void fraction. However, inferring the bubble-size distribution from void-fraction probes requires many problematic assumptions (S.G. Dias et al. 1997). For example, larger bubbles must not be distorted as they hit the probe; smaller bubbles must not bounce off the probe. In the laboratory, bubble sizing has generally relied on high-speed photography, beginning with Knapp and Hollander (1948). More recently, holography (Lauterborn and Hentschel 1985) and novel video techniques (Tassin and Nikitopoulos 1995; M.I. Dias et al. 1997) have been applied. Optical techniques are often unsuitable for industrial environments, where liquids may be opaque or visualization of the ¯ow impossible. Sensitive probes may be damaged in high-temperature or corrosive ¯ows. Even in laboratory models, the presence of particles in a bubbly ¯ow may preclude the use of optical techniques. Signi®cant voidage may also obscure regions of the ¯ow, making optical techniques impractical. The production of an acoustic signal by bubbles was ®rst detailed by Minnaert (1933). Bubbles produce an acoustic signal because of compression of the gas in the bubble. The compression may be caused by the recoil of the bubble neck on formation or by deformation of the bubble during its motion, although the precise mechanism of either process is still the subject of research (e.g., Manasseh et al. 1998). A full review of bubble acoustics is given by Leighton (1994). It was suggested by Leighton and Walton (1987) that the sound spectrum produced by bubbles in the environment could be used to calculate the bubble-size spectrum. Minnaert (1933) showed that under adiabatic conditions, the frequency of the acoustic signal is directly related to the bubble size according to:
v ! u u 3cP1 1 t f ; 2 R
2p q 0
1
where f is the frequency in Hz, P1 is the absolute liquid pressure, c is the ratio of speci®c heats for the gas, q is the liquid density and R0 is the bubble radius. It may seem surprising that surface tension is not involved, but it can be shown that this is a second-order effect (Longuet-Higgins et al. 1991). The bubble only rises a few We are grateful to Tracey Nicholls and Brook Dunstan for the bubble diameters while it emits the acoustic pulse. Hence photographic work and to numerous other colleagues at CSIRO P can be considered constant to within a few tenths of a 1 for help with setting up the acoustic instruments and electronics. percent. If the liquid is water and is not too deep, P1 is Knud Lunde (Norsk Hydro, Norway) was a great source of advice little changed from atmospheric pressure; for diatomic when setting up our ®rst bubble experiment. R. Manasseh (&), R. F. LaFontaine, J. Davy I. Shepherd, Y.-G. Zhu CSIRO Thermal and Fluids Engineering PO Box 56, Highett, VIC 3190 Melbourne, Australia
gases like air, the factor in Eq. (1) multiplying 1/R0 is roughly 3. Thus, bubbles 1 mm in radius emit a 3 kHz signal; bubbles 3 mm in radius emit a 1 kHz signal. These frequencies are generally much higher than those of mechanical and turbulent noises in a stirred-tank system. The existence of an acoustic signal leads to several potential experimental techniques. The `natural' acoustic emissions of a bubbly ¯ow can be used to deduce the bubble-size distribution. This could be called `passive' bubble sizing and is the subject of this paper. It is also possible to insonate a bubbly ¯ow and infer the bubble sizes. This could be called `active' bubble sizing and requires complex, specialized equipment and sophisticated analyses (Phelps and Leighton 1998). The passive acoustic signal can also be used as a trigger to enable accurate, high-resolution photography of the bubbles, providing a second check on the bubble size. Work to date on bubble acoustics has mostly investigated the signals produced by small, single bubbles. As bubbles rise they are distorted by hydrodynamic forces. These shape distortions alter the frequency. Strasberg (1953) has shown how the frequency may be corrected for prolate and oblate spheroidal bubbles. Furthermore, the shape distortions themselves induce acoustic oscillations by nonlinear parametric resonances (Longuet-Higgins 1989). As the air¯ow rate increases, the bubbling rate increases. The bubbling rate could be considered `high' once bubbles begin to collide. As the bubbling rate increases, bubbles also become larger and more distorted and begin to affect each other. Under these conditions, the relationship in Eq. (1) breaks down, and a small-amplitude correction to the basic frequency is no longer possible. Several studies have already been made of acoustic spectra in bubbly ¯ows. Pandit et al. (1992) took acoustic data from a turbulent jet containing bubbles. They related the sound-pressure power spectrum directly to the bubble-size distribution. The mean bubble size compared reasonably with photographic estimates. However, in estimating the bubble-size distributions, they relied on the assumption of uniform turbulence in the ¯ow exciting the bubble acoustic emissions. Their bubble size was in the range 0.1 to 1 mm in diameter. It was implied by Pandit et al. (1992) that they used a windowing technique, as is used in the present paper. Hsi et al. (1985) studied sound spectra in a stirred tank. They focused on lower frequencies (below 1 kHz). They associated this low-frequency sound with the formation of air cavities behind the impeller blades, the ¯ooding of the rotor with air and with large (tens of mm) bubbles. Boyd and Varley (1998) measured a broader spectrum at a point in a stirred tank and obtained a fair comparison between acoustically and photographically measured bubble sizes. Nevertheless, they noted that further work is required on the causes of bubble sound in this system before the technique can be universally applied. In particular, more fundamental work is required on bubble production under continuous sparging conditions. The bubble signals when bubbles are produced continuously appear different from those when the bubbles vibrate
in isolation. In the ®rst part of this paper, the signals of bubbles continuously produced from a nozzle are studied. In the second part, the ®ndings of the ®rst part are applied to the case of bubbles streaming off a ventilated cavity in a stirred tank.
2 Fundamental studies 2.1 Experimental method Bubble-acoustic calibrations on a single stream of continuously sparged bubbles were detailed elsewhere (Manasseh 1997) and only pertinent results are quoted here, together with additional optical data. Bubbles were produced from vertical-axis nozzles with internal diameters of 0.3, 0.5, 1.0, 2.0 and 4.0 mm, machined to maintain their internal edges as sharp as possible. This ensured a known contact radius for the forming bubble. The bubbles were produced in pressure-controlled mode, with the exception of the bubbles from the smallest nozzle, which were produced in volume-controlled mode. Explanations of these bubble-production modes are given in Chhabra (1993). Bubbles were produced at a depth of 0.238 0.0005 m in a box 23 cm square. A schematic of the equipment set-up is shown in Fig. 1. 2.2 Results A typical acoustic pulse is shown is Fig. 2. It was measured using a hydrophone (Bruel and Kjaer type 8103) near the bubble release point. Tests determined that the presence of the hydrophone did not alter the bubble dynamics or acoustic signal. The bubbling rate was 12 Hz, and the acoustic frequency was about 980 Hz. The corresponding spectrum, averaged over 30 bubbles, shows a clear peak at the bubble frequency (Fig. 3). The relevant part of the spectrum (below 2500 Hz) for a single bubble is virtually identical. After passing through ®lters (passband 600 Hz to 3 kHz) and a variable-delay trigger, the acoustic pulse was used to ®re a strobe that enabled high-resolution photographs to be taken using a 35 mm camera (Fig. 4). Such photographs, which are accurately related to the phase of the acoustic pulse, were used to calibrate the frequency-derived bubble sizes. To be compared precisely, the frequency measurement must be made at the same time as the optical measurement. A central issue is what kind of frequency measurement should be made. It can be seen that the pulse in Fig. 2 is modulated. The modulations become more marked as the bubbling rate increases. There is evidence that the modulations are associated with changes in bubble shape (Manasseh 1996). The modulations will, of course, introduce a lower frequency to the overall spectrum. Moreover, the basic frequency itself changes with time, probably because of the nonlinear factors mentioned in Sect. 1. Generally, the frequency decreases with time, each subsequent cycle being slightly longer. Hence, the frequency of the peak in a spectrum of the entire acoustic pulse, such as Fig. 3, will be lower than the frequency for the ®rst few
673
674
Fig. 1. Schematic of equipment set-up for continuous sparging tests
Fig. 2. Acoustic pulse from the formation of a single bubble
Fig. 3. Spectrum from the formation of single bubbles
periods of acoustic oscillation. As a result, the radius of a bubble assumed to be spherical and calculated using the spectral peak will be an overestimate of the bubble radius just after release. This effect can be seen by comparing Figs. 5 and 6. In Fig. 5, the frequency of the spectral peak is used to calculate the bubble radius via Eq. (1). In Fig. 6 the frequency used to calculate the bubble radius is in fact the reciprocal of the period of the ®rst acoustic oscillation. The physical processes associated with the ®rst acoustic cycle were studied by Manasseh et al. (1998), who found that the initial dip in sound pressure corresponds to the inrush of liquid as the neck joining a bubble to its parent body of gas begins to break. They also found that the ®rst peak corresponds to the compression of gas in the bubble by a jet penetrating the bubble after the neck breaks. Following these ®ndings, the ®rst period
is de®ned from the minimum in the initial dip in sound pressure to the minimum following the ®rst peak. For all but the highest air¯ow rates measured, the bubble production and acoustic oscillation are so regular that this ®rst period can be measured with an error less than 1%. It can be seen that the smaller bubbles are closest to ideal; for these, nonlinear errors due to distortions are minimal. The photographic measurements have random errors of up to 4% for smaller bubbles. Optical measurements are also subject to three systematic errors of interpretation. Firstly, it is never certain precisely where the bubble rim is. Secondly, systematic errors for the smaller nozzles (0.3 and 0.5 mm) may be up to 10%, owing to the use of the image of the nozzle tip as the datum scaling dimension in the photographs. It is, of course, the outer diameter of the
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Fig. 4. Bubble formation sequence
Fig. 5. Bubble radius from spectral-peak frequency R0sp
Fig. 6. Bubble radius from reciprocal of ®rst period R0fp
tip that appears in the image, not the precisely machined inner diameter. The outer diameter of the nozzle tip was measured using a graticule under a microscope but is still subject to an error of about 10%. Thirdly, although the bubble is nearly spherical at the instant the photograph was taken, there is still an eccentricity. Major and minor axes of the bubble were measured, and the photographic radii Rp were calculated as an equivalent spherical radius using: 676
Rp e1=3 dminor =2 ;
2
where e dmajor/dminor is the eccentricity and dmajor and dminor are the measured major and minor axis lengths. The eccentricity is at most 10% so this correction accounts for error of less than 3%. Thus, optical measurements should not be regarded as the `true' or `datum' bubble size; in effect, Fig. 6 is a comparison of two techniques for measuring bubble size, one optical and one acoustic. As a further check, bubblesizing software (Zhu et al. 1999) was run on digitized bubble images and compared with the acoustic measurements. Figure 7 shows that the software and the acoustic data agree reasonably well. For bubbles produced under precise laboratory conditions as described in Sect. 2.1, the bubbling rate ranges from about 0.5 to 50 Hz, and good agreement between the acoustic and photographic measurements is possible. Even at 50 Hz, a period of 20 ms ± some tens of cycles ± separates the acoustic pulses, so an overall spectral method should still be able to estimate the acoustic frequency. At a diffuser (for example, one used in the wastewater industry), pulses of sound may occur at over 100 Hz. In an industrial stirred tank such as the one described in Sect. 3.1, bubbles may stream off the impeller tips at 300±400 Hz, giving only one or two cycles of acoustic oscillation. Under these conditions, overall spectra tend to show peaks at the bubbling rate, not at the bubble size. Away from the impeller
Fig. 7. Acoustic versus digitized optical radii
tips in a full-scale tank, sound pulses occur at 3±10 Hz, well within the range of the present experiments. The bubble ®rst-period frequency was used to calculate the bubble volume using Eq. (1), which, together with the bubbling rate, can be used to reconstruct the air¯ow rate through the nozzle. Comparing this with the known ¯ow rate obtained from a calibration gives a useful check of the accuracy of the acoustic technique. This is shown in Fig. 8. The agreement on air¯ow rate appears better than that on bubble radius alone. This is because the air¯ow rate is the product of bubble volume and bubbling rate. As the air¯ow rate increases, the bubbling rate increases much more than bubble volume; hence errors on bubble volume become proportionately smaller. As before, the volume is calculated as 4/3pr3e, where e is the eccentricity and r is the minor-axis radius, using estimates of the bubble eccentricity gained from photographic work. The agreement is good and linear up to a limit that corresponds to the beginning of bubble pairing. Here, the bubble just formed collides with the bubble immediately above (Manasseh 1996). Under some circumstances the bubbles can coalesce; however, in this experiment surface tension is suf®ciently low to prevent coalescence. The implication of this work is that for estimates of bubble size, individual bubble pulses should be stored and analysed separately, rather than taking overall spectra. In particular, the earliest cycles of the bubble pulse will yield the most accurate data.
3 Tests on a stirred, sparged tank 3.1 Equipment, tests and procedure Tests of the bubble-acoustic technique were performed on a model of an actual tank used in industry. (It was a 1: 12.9 scale model of a tank from the Becher process used in the
Fig. 8. Air¯ow rate reconstructed
production of titanium dioxide.) Most aeration tanks used in industry have a similar schematic form: a cylindrical tank, stirred by an impeller on the centreline and close to the bottom, and ®tted with four vertical baf¯es to prevent swirling of the bulk of the ¯ow. The same generic aeration tank is used in many chemical and minerals plants and also in biotechnology (cell-culture) reactors. The tank held 20 l of tap water; its diameter was 290 mm. It was made of clear acrylic and was surrounded by a clear water-®lled box, thus enabling some visualization of the ¯ow. The turbine speed and air¯ow rate were scaled to give the same pattern of gas distribution as in the full-scale vessel, while maintaining the Reynolds number in the turbulent range and achieving Froude number similarity for the surface motion. Evidence that this scaling was successful was that the holdup, or fractional increase in liquid level owing to aeration, was the same (about 10%) in the model as in the full scale tank. The speed of the impeller, a six-bladed Rushton turbine of diameter D 131 mm, was held at 250 rpm. The Rushton turbine is basically a disc with a number of vertical blades ®xed normal to it. Air is sparged in below the turbine disc and hence gets distributed by the turbine's action. The principle of the Rushton turbine is the formation of low-pressure regions in the trailing vortices behind each blade. These low-pressure regions attract air into them and form cavities that shed continuous streams of bubbles. An air¯ow rate of Q 1.6 l s)1 was used. The ¯uid height was 267 mm. For these conditions, the bubbles form structures known as `vortex cavities' behind each blade of the Rushton turbine (Warmoeskerken et al. 1984). To facilitate measurements at many points in the tank, the probe pivoted about a vertically movable point, while the tank could be rotated 90° about its axis. Probe and tank motion were controlled by a computer, as was data recording. The work reported in the Sect. 2 indicated that only the earliest stages of each acoustic pulse yield accurate information on the bubble size. Moreover, when dealing with a complex sound spectrum, some method of `windowing' the relevant signals out of the noise is required. Any such method is imperfect and introduces some skew into the results. The following windowing procedure was adopted:
time tstore required to store one pulse was about 1.97 s, much longer than the 15 ms capture time. 3. The system was left to capture n 100 pulses. These were stored individually, together with data on the time it took to capture each pulse. A similar windowing technique was used by Pandit et al. (1992). Although they did not explicitly state why, it was probably the most practical way of analysing the data, just as with the present data. As a test, a spectrum was obtained when the data were analysed continuously, without determining if a bubble pulse was present or not. The intention was to compare this spectrum with that obtained using the triggering method. The signal was low-pass ®ltered at the charge ampli®er with a 10 kHz cut-off and then high-pass ®ltered with a 500 Hz cut-off. To ensure exactly the same data were analysed by the two techniques, it was recorded on Digital Audio Tape (DAT) by a Sony TCD D7 recorder that has a digitising rate of 44 kHz, and this data was played back through the two techniques. Figure 9 shows the acoustic spectrum at one point in the tank, when the data were analysed continuously. A digital oscilloscope (HP 35670A) was set to 1,600 lines resolution (4,096 points) and a span of 12.8 kHz, which corresponds to a sample length of 125 ms. Hence 2,224 averaged samples of this data cover 278 s. These 2,224 samples took 280 s; thus, the data coverage was effectively continuous. Only the part of the spectrum below 3 kHz is shown in Fig. 9 since there is no signi®cant signal power above 3 kHz. Checks for aliasing were performed by resampling the data at up to 104 kHz. The mean of the 2,224 averaged spectra is the central curve in Fig. 9; the two bounding curves represent the 95% statistical con®dence limits. The sharp drop-off in signal power at 500 Hz is due to the high-pass ®ltering applied to the signal. The signal power falls quickly until about 2 kHz, and it tails off gradually above 2.5 kHz.
1. The signal was observed with various trigger settings and at various probe locations. By trial and error, a trigger level was established manually, such that only pulses corresponding to clear bubble signals were captured. For most of the experiments reported here, this trigger level corresponded to a sound pressure of 186.2 Pa at the hydrophone location. 2. This trigger level was then set, together with a capture time of 15 ms, on the digital oscilloscope, which was controlled by the computer. This capture time was enough to obtain several cycles of acoustic oscillation after the initial sharp rise of the pulse, but no more. Because the data were acquired by the oscilloscope, then transferred over a serial line to the computer, the minimum Fig. 9. Continuous spectrum from a stirred, sparged tank
677
to obtain 100 pulses, while close to the turbine the time to obtain 100 pulses tended to the minimum of 1.97 s. Data recording commenced with the probe pivoted to the setting closest to vertical and the probe at the lowest sampling point. The probe was then progressively raised through the higher levels, waiting at each until all the required data had been obtained. After the probe recorded the upper-most position, the tank was rotated 5° and the above sequence was repeated. 678
Fig. 10. Windowed spectrum from a stirred, sparged tank
3.2 Results ± spatial distribution of bubble sizes Figures 11, 12 and 13 show results of the acoustic technique in the model stirred, sparged tank. The plane shown cuts a diameter through the tank intersecting a pair of baf¯es. The probe-traversing system described in Sect. 3.1 resulted in data on a grid of 14 points across the diameter and 13 points in the vertical; there were 7 such planes cutting progressively smaller chords of the tank, so the plane shown is the one with the most data. The data have been bilinearly interpolated within the pixel corresponding to each measurement location to give a continuous effect, but no additional data smoothing or spatial ®ltering has been performed. The signi®cant asymmetry is due to the presence of baf¯es in the tank coupled with the swirling nature of the ¯ow. The impeller was turning clockwise, viewed from above. Figure 11 shows the distribution of average bubble sizes, obtained through a windowed spectrum, calculated in the same way as that of Fig. 10. In fact, Fig. 10 shows the data from the point at the bottom row of the tank and just to the left of the white `danger zone' surrounding the impeller in Fig. 11. Recall that Fig. 10 shows the average of 100 windowed spectra. To get an average bubble size at each point, the mean frequency of spectra such as that of
The spectrum obtained by the windowing technique using the same (HP 35670A) digital oscilloscope is shown in Fig. 10. Here a trigger was set with a level corresponding to a sound pressure of 186.2 Pa at the hydrophone location (roughly 40% of the typical peak signal). Data were captured for 15.6 ms and zero-padded out to 125 ms. The spectra were taken at the same resolution (1,600 lines) as for the continuous analysis, giving the same frequency span of 12.8 kHz. The spectra from 100 windowed samples were averaged. Figure 10 represents considerably less data than Fig. 9 (100 samples versus 2,224 samples); therefore the 95% con®dence limits bounding the averaged curve are broader. It is clear that the windowing technique results in a much peakier spectrum. Individual sub-peaks that may correspond to features of the bubble-size distribution are apparent. This information has been lost from the continuous spectrum. The windowing technique focuses the analysis on the ®rst few cycles after a peak in signal intensity. Since the loudest sounds in this system are made by bubbles, the windowing technique focuses the analysis on the ®rst few cycles of bubble-acoustic oscillation, which from Sect. 2.2 gives the best estimate of bubble size. Clearly, however, a window selected by a signal-level trigger biases the analysis towards bubbles that are large and very close to the hydrophone, because the sound pressure falls off both with distance and bubble size. It also biases the analysis towards bubbles excited to a higher amplitude; this will be discussed in Sect. 3.3. The same trigger level was used both near to and far away from the turbine. Close to the turbine-blade tips, bubbles are produced continually from the cavities behind the blades and loud bubble pulses occur virtually continuously. The trigger level setting was such that, at these locations, the time spent waiting for a suitably loud bubble pulse was virtually zero. However, in regions of the tank far from the bubbleformation zones, bubble pulses loud enough to be captured Fig. 11. Distribution of average bubble size, fast Fourier transwere rare. In some locations, it took the system half an hour form method
In both Figs. 11 and 12, the zone of largest bubbles is found above the impeller, and a `chimney' of large bubbles rises up the tank centre to the surface. This can be explained, given an understanding of the typical ¯ow in such tanks. The impeller creates a toroidal vortex that sends small, newly-formed bubbles radially outwards. On reaching the wall, some bubbles are recirculated inwards and downwards. Because they are moving against their buoyancy, these bubbles are the slowest moving and thus have the greatest opportunity to coalesce (Bhaga and Weber 1981). The result is a zone of large bubbles above the impeller. These large bubbles are sucked into the impeller blades, forming the two `legs' in the pattern (really a cone in three dimensions).
3.3 Estimate of void-fraction distribution While the spatial distribution of bubble size is important, it is equally important for mass-transfer applications to know the distribution of void fraction or the population density of bubbles. The acoustic technique cannot directly provide a void fraction; however, it may be possible to Fig. 12. Distribution of average bubble size, ®rst period method estimate the void fraction, employing some assumptions based on the use of the windowing technique. It is worth noting in passing that other acoustic techniques may be able to measure void fraction directly (e.g., an active measurement of the speed of sound). It can be shown that instantaneous sound pressure p(t) produced by a single bubble is given by p
t2
1 3cP31 =
4p2 q X
t2 ; f 2
c
c 1r 2
3
where the bubble is undergoing adiabatic compressions, f is the bubble's natural frequency given by Eq. (1), r is the distance from the bubble and the time-dependent factor X(t) is given by
4 X
t 3
R0 3c 1 1 R0 2 c ; R
t 3 R
t
4
(Pandit et al. 1992), where R(t) is the instantaneous value of the bubble's radius as it oscillates about its equilibrium radius R0. Unfortunately, we do not know what the function R0/R(t) is in general; this requires knowledge of the phenomena distorting the bubble, which may well be different for different sized bubbles. In a generally turbulent ¯ow Fig. 13. Acoustically estimated void fraction, / (percent) where energy is present at a wide range of length scales, it Fig. 10 were calculated, and the bubble size calculated may be valid to assume that R0/R(t) is the same for all from this frequency simply by using Eq. (1). It is also bubble sizes; this was assumed by Pandit et al. (1992) for possible to obtain a distribution of bubble sizes at each their turbulent jet. It is unlikely that we can make the same point, following a procedure such as that described by assumption in all regions of the more complex stirred Pandit et al. (1992), but the aim here is simply to see how tank. Nevertheless, near the impeller tips, the sounds rethe bubble size varies with location in the tank. corded are those of bubbles being formed. Bubble forFigure 12 shows the distribution of average bubble sizes mation causes distortions to the bubble shape that are selfcalculated from the ®rst period alone, as described in similar for bubbles 2±8 mm in diameter (Manasseh 1997), Sect. 2.2. The general pattern is very similar; only, as ex- so there may be some validity for the assumption of selfpected from the results of Sect. 2.2, the bubble sizes are similar R0/R(t) in this zone. somewhat smaller. The similarity of the pattern is evidence In a system where R0/R(t) is self-similar, the peak value that the bubble-acoustic technique is quite robust: quite of X(t) is independent of bubble size, so the peak value p^ of different analysis techniques give similar results. p(t) is given from Eq. (3) by
679
2 p^ kR20 =r2 ;
5
where k is a constant, since 1/f 2 / R20 from Eq. (1). When individual bubble pulses are captured with the use of a ®xed trigger level (hence a ®xed p^, say ptrig), a bubble of given size R0 will be detected by the system provided its centre is within a critical radius rc of the hydrophone, given by
680
p k rc R0 : ptrig
6
In other words, if we accept the assumption of self-similar bubble distortions (`Assumption 1'), a bubble twice as big will be detected twice as far away for the same trigger level. Next, assume that the windowing period is suf®ciently brief that overlap events do not occur (`Assumption 2'). In other words, the actual bubble detected is the only bubble within the critical radius during the windowing period. The probability that two bubbles coexist within the critical radius during the windowing period is considered negligible. Therefore, the instantaneous void fraction a (during the windowing period) is given by the ratio of the bubble volume to the volume within the critical radius (plus, to be precise, one bubble radius),
a
R30 ;
rc R0 3
7
Sect. 2.2 for bubbles of the size range found in the stirred tank. For the present paper the intention is not to accurately quantify a, but to use the fact that a may be considered constant to determine the relative spatial distribution of void fraction. It should be noted in passing that for some experiments, the trigger level was set suf®ciently high that a 0.13 or roughly 1/8; thus, rc.R0 and the bubble had to be virtually touching the hydrophone for it to be detected. In this case, any error in assuming the bubble detected is the only bubble within the critical radius is virtually eliminated. (Although to be precise when such close-range measurements are being made, the volume of the hydrophone itself should be included.) Hence, by estimating the `instantaneous' void fraction in this way, it is possible to make the hydrophone function virtually as an electrical or optical void-fraction probe ± without having to pierce the bubble. Of course, because the volume of the hydrophone (a cylinder about 25 mm long and 9.5 mm in diameter) is much greater than the volume of the tip of a typical void-fraction probe, the spatial resolution is much poorer, but is still good for testing most industrial systems. For most of the present experiments, a was 0.39. To estimate the average void fraction, it remains to calculate the fraction of the total time that bubbles are present within the critical radius. Since, with the present system, a ®nite time is required to detect a bubble pulse, the average void fraction / is given by
neglecting the volume occupied by the hydrophone itself. ntstore /a ;
8 Since from Eq. (6) rc / R0, the conclusion is that the ttotal instantaneous void fraction a is a constant. Rather than guessing the constant k, which depends on the unknown which makes the assumption that if the time taken to collect all the pulses (ttotal) is equal to the minimum possible R0/R(t), it was determined from the experiments of
Fig. 14. Void-fraction distribution measured by conductivity probe
(ntstore), the average void fraction is the maximum possible (`Assumption 3¢). The result is shown in Fig. 13. It must be emphasized that this void fraction distribution should only be used qualitatively. The distribution of estimated void fraction is based on three assumptions. Of these, Assumption 1 ± that bubble distortions are selfsimilar ± is questionable away from the bubble-formation zones; and Assumption 2 ± that bubble detections do not overlap ± becomes contentious the longer the windowing period and the lower the trigger level. Nonetheless, the resulting distribution is generally consistent with both visual observations and data from an electrical void-fraction probe. The highest void fraction is in the region near the impeller tips. Considering both Figs. 12 and 13, it may be concluded that the tank is performing reasonably well: the region of high void fraction generally contains smaller bubbles. There is an interesting discrepancy between the acoustically estimated void fraction and the void fraction measured by an electrical probe (Fig. 14). The electric conductivity probe effectively averages the time its tip spends in air, and although this technique also has its problems (S.G. Dias et al. 1997), it is likely to be more reliable than the acoustic void fraction, given the assumptions above. The signi®cant discrepancy is found at the edge where the tank ¯oor and wall join. Here, the acoustic method reports a relatively high void fraction but the electrical probe reports a relatively low void fraction. One possible explanation is an acoustic ampli®cation at walls and edges, but many other effects could be responsible.
4 Conclusion An acoustic technique for bubble sizing can be applied at practical bubbling rates. However, nonlinear effects during the bubble rise alter the frequency. A spectral measure of bubble frequency can be related to the radius of the bubble just on formation, provided allowances are made for the frequency changes. The best match to the photographically measured bubble radius is obtained when a frequency based on the ®rst period of acoustic oscillation is used. These results implied that the best technique was to capture individual bubble pulses and analyse them separately, rather than measuring overall spectra. Moreover, only the ®rst few cycles of the pulses should be analysed. Tests conducted on a stirred, sparged tank obtained a realistic distribution of mean bubble size with location in the tank. Further comparisons with an optical technique are recommended to cross-check this distribution. Reliable data are rapidly gathered near where bubbles are being formed and where the population density of bubbles is high, while reliable data rates are low in regions of low bubble population remote from the bubble formation. This is the reverse of optical methods, which fail in high bubble concentrations and excel in low concentrations. It is also possible to obtain a rough distribution of average void fraction using the acoustic technique. However, three assumptions must be made to obtain a void-fraction distribution in a stirred tank environment. In particular, self-similar bubble distortions must be assumed, which is questionable in a complex bubbly ¯ow. Assumptions on
the lack of event overlap within a window and the maximum void fraction are also questionable. Further research is also required to see how the passive acoustic technique can be translated into a practical measurement tool in a complex bubbly ¯ow. In particular, the mechanism by which bubbles produce sound remote from their formation zone needs to be understood. Nevertheless, in many practical industrial systems, the need is not to accurately measure bubble size distributions, but to measure a relative change over time of mean bubble size. This is because the need is for feedback control of aeration. The bubble-size measurement protocol detailed in this paper has recently been successfully applied by the authors over a 24-h period in a working 20,000 l bioreactor. In a simpler bubbly ¯ow where spatial distributions of the bubble size and void fraction are determined at their formation points, such as in a sparging plate or a diffuser, absolute (rather than relative) measurements of bubble size distributions should also be possible.
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